Matrix and key generation device, matrix and key generation system, matrix coupling device, matrix and key generation method, and program

ABSTRACT

A vector generation unit generates a vector x so that xn[i]≠xn[j] if kn[i]=k[j] at i≠j. A set generation unit generates a set Bn,j so that individual elements correspond to combinations of the N−1 pieces of elements, which are individually selected from sets M0, . . . , MN−1 other than a set Mn, and xn[j] and the elements for all of the combinations are included. A matrix generation unit generates a matrix Tn′ so that the matrix Tn′ includes rows identical to Tn[j] in the number equal to the number of elements of the set Bn,j. A key generation unit generates a vector kn′ so that elements of the matrix Tn′ which correspond to a row identical to Tn[j] correspond to combinations of kn[j] and elements of the set Bn,j and further, the elements of the set Bn,j are different from each other when there are a plurality of rows identical to Tn[j].

TECHNICAL FIELD

The present invention relates to a matrix and key generation device, a matrix and key generation system, a matrix coupling device, a matrix and key generation method, and a program for processing data constituted of rows and columns of a table or the like.

BACKGROUND ART

Non-patent Literature 1 discloses a technique in which sorting of orders of rows of a matrix and sorting of orders of elements of a vector are performed by secure computation, with respect to the matrix of which elements are concealed and vectors which corresponds to each row of the matrix and includes concealed elements, in accordance with the elements of the vectors while keeping the elements concealed. Non-patent Literature 2 discloses a technique in which computation for counting the number of times of appearance (the number of pieces of duplication) of an identical element (referred to below as “step computation”) is performed while keeping the elements concealed and concealed results are outputted. FIG. 1 illustrates a general outline of the step computation. Here, ∥ ∥ is a symbol representing a concealed state. Since ∥1∥ which is the first element of the input appears for the first time, the first element of the output is ∥1∥. Since ∥2∥ which is the second element of the input appears for the first time, the second element of the output is ∥1∥. Since ∥2∥ which is the third element of the input appears for the second time, the third element of the output is ∥2∥. Since ∥3∥ which is the sixth element of the input appears for the third time, the sixth element of the output is ∥3∥.Thus, through the performance of the step computation, the number of times of appearance of an identical element (the number of pieces of duplication) is counted and a result thereof is outputted. Non-patent Literature 3 discloses a technique in which a plurality of matrices of which elements are concealed are coupled based on concealed elements of vectors corresponding to respective matrices with a small amount of communication. Non-patent Literature 4 discloses a technique for concealing data (distribution processing), a technique for reconstructing concealed data (reconstruction processing), a technique in which addition, multiplication, and verification (verification of whether two concealed data are equal to each other) are performed in a state that the concealed data are kept concealed so as to obtain concealed results, and the like.

PRIOR ART LITERATURE Non-Patent Literature

Non-patent Literature 1: Koki Hamada, Dai Ikarashi, Koji Chida, and Katsumi Takahashi, “Oblivious radix sort: An efficient sorting algorithm for practical secure multi−party computation”, IACR Cryptology ePrint Archive, Vol. 2014, p. 121, 2014.

Non-patent Literature 2: Koki Hamada, Dai Ikarashi, and Koji Chida, “An Algorithm for Computing Aggregate Median on Secure Function Evaluation”, In CSS, 2012.

Non-patent Literature 3: Koki Hamada, Ryo Kikuchi, Dai Ikarashi, and Koji Chida, “An Equijoin Algorithm on Secure Function Evaluation”, the collection of papers of the 26th Annual Conference of the Japanese Society for Artificial Intelligence, June 2012.

Non-patent Literature 4: Koji Chida, Koki Hamada, Dai Ikarashi, and Katsumi Takahashi, “A Three-party Secure Function Evaluation with Lightweight Verifiability Revisited”, In CSS, 2010.

SUMMARY OF THE INVENTION Problems to be Solved by the Invention

The technique for coupling matrices described in Non-patent Literature 3 realizes coupling of matrices through secure computation with a small amount of communication on the assumption that there is no duplication among elements of a vector which is a key of the coupling. However, this technique cannot be used disadvantageously in the case where there is duplication among elements of a vector. In order to make the technique for coupling matrices of Non-patent Literature 3 applicable, it is necessary to convert a vector which includes duplicate elements and a matrix which is a coupling object respectively into a vector which includes no duplicate elements and a matrix corresponding to the vector.

An object of the present invention is to provide a technique for converting a vector which includes duplicate elements and a matrix which is a coupling object respectively into a vector which includes no duplicate elements and a matrix corresponding to the vector.

Means to Solve the Problems

First, it is assumed that N is an integer which is 1 or larger, n is an integer which is between 0 and N−1 inclusive, K_(n), is an integer which is 1 or larger, i and j are integers, T_(n), is a matrix having K_(n), rows, k_(n), is a vector which includes K_(n) pieces of elements, T_(n)[j] is a row on a j-th order of the matrix T_(n), k_(n),[j] is an element on a j-th order of the vector k_(n), and m_(n) is an upper limit number in which the elements of the vector k_(n), are duplicated. It is assumed that the element on the j-th order of the vector k_(n) is an element corresponding to a row on the j-th order of the matrix T_(n).

The first invention of the present invention is a technique which can be combined with the matrix coupling described in Non-patent Literature 3 and is limited to secure computation. In the first invention, it is assumed that ∥ ∥ is a sign representing concealed data and M_(n) is a set whose elements are ∥1∥, . . . , ∥m_(n)∥. The matrix and key generation device according to the first invention constitutes a matrix and key generation system by three or more matrix and key generation devices which are mutually connected via a network. In the matrix and key generation system, secure computation can be performed among the matrix and key generation devices while performing data communication. Further, the numbers of rows and columns of the matrix T, the number of elements of the vector k_(n)and the value m are unconcealed information and each element of the matrix T_(n), and each element of the vector k_(n) are concealed information among the matrix and key generation devices. The matrix and key generation device includes a sort unit, a vector generation unit, a set generation unit, a matrix generation unit, and a key generation unit.

The sort unit performs sorting of orders of rows of the matrix T_(n) and sorting of orders of elements of the vector k_(n), with respect to each of n=0, . . . , N−1 through secure computation with sort units of other matrix and key generation devices in accordance with the elements of the vector k_(n), while maintaining correspondence, so as to update matrices T₀, . . . , T_(N−1) and vectors k₀, . . . , k_(N−1) with the matrices and the vectors after the sorting. The vector generation unit, the set generation unit, the matrix generation unit, and the key generation unit perform processing with respect to the matrices T₀, . . . , T_(N−1) and the vectors k₀, . . . , k_(N−1) which are updated by the sort unit. The vector generation unit generates a vector x_(n), of which a number of pieces of elements is K_(n), and each of the elements is concealed, with respect to each of n=0, . . . , N−1 through secure computation with vector generation units of other matrix and key generation devices so that x_(n)[1]=∥1∥ and x_(n)[i]=∥x_(n)[i−1]+1∥ if k_(n)[i−1]=k[i] and x_(n)[i]=∥1∥ if k_(n)[i−1]≠k_(n)[i] with respect to 2≤i≤K_(n). The set generation unit generates a set B_(n) of which each element is concealed, with respect to each of n=0, . . . , N−1 and each of j=1, . . . , K_(n) through secure computation with set generation units of other matrix and key generation devices so that individual elements correspond to combinations of N−1 pieces of elements, the N−1 pieces of elements being individually selected from sets M₀, . . . , M_(N−1) other than the set M_(n), and x_(n)[j], and the elements for all of the combinations are included. The matrix generation unit generates a matrix T_(n)′, of which each element is concealed, with respect to each of n=0, . . . , N−1 through secure computation with matrix generation units of other matrix and key generation devices, so that rows identical to T_(n)[j] are included in a number equal to a number of elements of the set B_(n,j) for all of j=1, K_(n). The key generation unit generates a vector k_(n)′, of which each element is concealed, with respect to each of n=0, . . . , N−1 through secure computation with key generation units of other matrix and key generation devices, so that an element of the matrix T_(n)′, the element corresponding to a row identical to T_(n)[j], corresponds to a combination of k_(n)[j] and an element of the set B_(n), and further, the elements of the set B_(n,j) are different from each other when there are a plurality of rows identical to T_(n)[j].

The second invention of the present invention is a technique for converting a vector which includes duplicate elements and a matrix which is a coupling object respectively into a vector which includes no duplicate elements and a matrix corresponding to the vector without performing secure computation. In the second invention, it is assumed that M_(n) is a set composed of m_(n) pieces of elements which are different from each other and M_(n)[i] is an element on the i-th order of the set M_(n). Further, it is assumed that matrices T₀, . . . , T_(N−1) and vectors k₀, k_(N−1) are inputs. The matrix and key generation device according to the second invention includes a vector generation unit, a set generation unit, a matrix generation unit, and a key generation unit. The vector generation unit generates a vector x_(n), of which a number of pieces of elements is K_(n) and each of the elements is an element of the set M_(n), with respect to each of n=0, . . . , N−1 so that x_(n)[i]≠x_(n)[j] if k_(n)[i]=k_(n)[j] at i≠j. The set generation unit generates a set B_(n,j) with respect to each of n=0, . . . , N−1 and each of j=1, . . . , K_(n) so that individual elements correspond to combinations of N−1 pieces of elements, the N−1 pieces of elements being individually selected from sets M₀, . . . , M_(N−1) other than the set M_(n), and x_(n)[j] and the elements for all of the combinations are included. The matrix generation unit generates a matrix T_(n)′ with respect to each of n=0, . . . , N−1 so that rows identical to T_(n)[j] are included in a number equal to a number of elements of the set B_(n,j) for all of j=1, . . . , K_(n). The key generation unit generates a vector k_(n)+ with respect to each of n=0, . . . , N−1 so that an element of the matrix T_(n)′, the element corresponding to a row identical to T_(n)[j], corresponds to a combination of k_(n)[j] and an element of the set B_(n,j) and further, the elements of the set B_(n,j) are different from each other when there are a plurality of rows identical to T_(n)[j].

Effects of the Invention

According to the matrix and key generation system and the matrix and key generation device of the present invention, a vector which includes duplicate elements and a matrix which is a coupling object can be respectively converted into a vector which includes no duplicate elements and a matrix corresponding to the vector.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a general outline of step computation.

FIG. 2 illustrates a specific example of coupling of matrices.

FIG. 3 illustrates a configuration example of a matrix and key generation system according to the first embodiment.

FIG. 4 illustrates a processing flow of the matrix and key generation system according to the first embodiment and a processing flow of a matrix and key generation device according to the second embodiment.

FIG. 5 illustrates a state that sorting of orders of rows of the matrix T₀ and sorting of orders of elements of the vector k₀ illustrated in FIG. 2 are performed in accordance with the elements of the vector k₀ through secure computation while maintaining correspondence.

FIG. 6 illustrates vectors x₀, x₁, and x₂ which are results of step computation performed with respect to vectors k₀, k₁, and k₂ which are illustrated in FIG. 2 and are in the state that elements are concealed.

FIG. 7 illustrates examples of the matrix T₀′, the vector k₀′, the matrix T₁′, and the vector k₁′ in the state that elements are concealed in the case of coupling the matrix T₀ and the matrix T₁ illustrated in FIG. 2.

FIG. 8 illustrates a configuration example of a matrix coupling system according to a modification of the first embodiment.

FIG. 9 illustrates a processing flow of the matrix coupling system according to the modification of the first embodiment.

FIG. 10 illustrates a matrix obtained by coupling the matrix T₀′ and the matrix T₁′ which are illustrated in FIG. 7.

FIG. 11 illustrates a configuration example of a matrix and key generation device according to the second embodiment and a modification of the second embodiment.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Embodiments according to the present invention will be detailed below. Components having identical functions will be denoted by identical reference numerals and duplicate description will be omitted.

First Embodiment

In the description of the first embodiment, it is assumed that N is an integer and 1≤N, n is an integer and 0≤n≤N−1, K_(n,j) is an integer and 1≤K_(n), i and j are integers, T is a matrix having K_(n) rows, k_(n) is a vector which includes K pieces of elements, T_(n)[j] is a row on the j-th order of the matrix T_(n), k_(n)[j] is an element on the j-th order of the vector k_(n), m_(n) is an upper limit number in which the elements of the vector k are duplicated, M_(n) is a set whose elements are 1, . . . , m_(n), and ∥ ∥ is a sign representing concealed data. The j-th element of the vector k is an element corresponding to the j-th row of the matrix T. Here, the upper limit number m_(n), does not have to be a number in which elements of the vector k_(n) are actually duplicated but may be set to the maximum value in which elements are potentially duplicated.

<Coupling of Matrices>

A matrix expressing a table, a column vector expressing a key of each row of the matrix, and coupling of matrices will be first described. In the coupling of matrices, in the case where there is an element common to all of vectors k₀, . . . , k_(N−1), rows, which correspond to the common element, of matrices T₀, . . . , T_(N−1) are coupled to obtain one row. FIG. 2 illustrates a specific example of coupling of matrices. The matrices T₀, T₁, and T₂ are coupling objects. The vectors k₀, k₁, and k₂ are column vectors representing keys. Here, in this application, “matrix” represents a form for expressing a table and “vector” represents a form for expressing a key, so that each element of the matrix T_(n) and the vector k_(n) does not have to be limited to one numerical value but may be a combination of numerical values, a character string, or the like.

Here, coupling between the matrix T₀ and the matrix T₁ will be considered. The first and fourth elements of the vector k₀ representing keys of the matrix T₀ and the first and second elements of the vector k₁ representing keys of the matrix T₁ are “1”, being mutually common. That is, the keys on the first and fourth rows of the matrix T₀ and the keys of the first and second rows of the matrix T₁ are “1”, being mutually common. Further, the key on the third row of the matrix T₀ (the third element of the vector k₀) and the keys on the third and fourth rows of the matrix T₁ (the third and fourth elements of the vector k₁) are “3”, being mutually common. Accordingly, in a matrix obtained by coupling the matrix T₀ and the matrix T₁, a result obtained by coupling the first row of the matrix T₀ and the first row of the matrix T₁ is on the first row, a result obtained by coupling the first row of the matrix T₀ and the second row of the matrix T₁ is on the second row, a result obtained by coupling the fourth row of the matrix T₀ and the first row of the matrix T₁ is on the third row, a result obtained by coupling the fourth row of the matrix T₀ and the second row of the matrix T₁ is on the fourth row, a result obtained by coupling the third row of the matrix T₀ and the third row of the matrix T₁ is on the fifth row, and a result obtained by coupling the third row of the matrix T₀ and the fourth row of the matrix T₁ is on the sixth row.

Subsequently, coupling of the matrix T₀, the matrix T₁, and the matrix T₂ will be considered. Among elements of the vector k₀ representing keys, elements of the vector k₁ representing keys, and elements of the vector k₂ representing keys, the first and fourth elements of the vector k₀, the first and second elements of the vector k₁, and the first element of the vector k₂ are “1”, being mutually common. That is, the keys on the first and fourth rows of the matrix T₀, the keys on the first and second rows of the matrix T₁, and the keys on the first row of the matrix T₂ are “1”, being mutually common. Accordingly, in a matrix obtained by coupling the matrix T₀, the matrix T₁, and the matrix T₂, a result obtained by coupling the first row of the matrix T₀, the first row of the matrix T₁, and the first row of the matrix T₂ is on the first row, a result obtained by coupling the first row of the matrix T₀, the second row of the matrix T₁, and the first row of the matrix T₂ is on the second row, a result obtained by coupling the fourth row of the matrix T₀, the first row of the matrix T₁, and the first row of the matrix T₂ is on the third row, and a result obtained by coupling the fourth row of the matrix T₀, the second row of the matrix T₁, and the first row of the matrix T₂ is on the fourth row.

<Configuration and Algorithm>

FIG. 3 illustrates a configuration example of a matrix and key generation system and FIG. 4 illustrates a processing flow of the matrix and key generation system. The matrix and key generation system includes three or more matrix and key generation devices 160 which are mutually connected via a network 800 and secure computation can be performed among these matrix and key generation devices 160. Each of the matrix and key generation devices 160 includes a sort unit 110, a vector generation unit 120, a set generation unit 130, a matrix generation unit 140, a key generation unit 150, and a recording unit 190. In the present embodiment, the numbers of rows and columns of the matrix T_(n), the number of elements of the vector k_(n)and the value m_(n), are unconcealed information and each of the elements of the matrix T_(n), and each of the elements of the vector k_(n), are concealed information among the matrix and key generation devices 160. That is, shares of respective elements of the matrix T_(n), and shares of respective elements of the vector k_(n), are distributed and recorded in the recording units 190 of a plurality of matrix and key generation devices. Here, a “share” is data with which an original value can be reconstructed when a predetermined number of shares are known and is called “distributed data” in Non-patent Literature 4.

The sort unit 110 performs sorting of orders of rows of the matrix T_(n) and sorting of orders of elements of the vector k_(n), with respect to each of n=0, . . . , N−1 through secure computation with the sort units 110 of other matrix and key generation devices 160 in accordance with the elements of the vector k_(n), while maintaining correspondence, so as to update the matrices T₀, . . . , T_(N−1) and the vectors k₀, . . . , k_(N−1) with the matrices and the vectors after the sorting (S110). A method for performing the sort processing through the secure computation is specifically described in Non-patent Literature 1. Further, sorting may be performed in an ascending order or a descending order. FIG. 5 illustrates a state that sorting is performed with respect to orders of rows of the matrix T₀ and orders of elements of the vector k₀ illustrated in FIG. 2 through secure computation in accordance with the elements of the vector k₀ while maintaining correspondence. In this example, keys (elements of the vector k₀) having smaller values are on younger orders (upper). The vector generation unit 120, the set generation unit 130, the matrix generation unit 140, and the key generation unit 150 perform processing with respect to the matrices T₀, . . . , T_(N−1) and the vectors k₀, . . . , k_(N−1) which are updated by the sort unit 110.

The vector generation unit 120 generates a vector x_(n), of which the number of pieces of elements is K_(n) and each of the elements is concealed, with respect to each of n=0, . . . , N−1 through secure computation with the vector generation units 120 of other matrix and key generation devices 160 so that x_(n)[1]=∥1∥ and x_(n)[i]=∥x_(n)[i−1]+1∥ if k_(n)[i−1]=k_(n)[i] and x_(n)[i]=∥1∥ if k_(n)[i−1]≠k_(n)[i] with respect to 2≤i≤K_(n) (S120). This processing is same as the step computation illustrated in Non-patent Literature 2. Further, the technique for concealing values is described in Non-patent Literature 4 and the like. FIG. 6 illustrates vectors k₀, k₁, and k₂ of which elements are concealed and which are obtained by performing sorting by the sort unit 110 with respect to the vectors k₀, k₁, and k₂ illustrated in FIG. 2, and vectors x₀, x₁, and x₂ which are results of the step computation performed, by the vector generation unit 120, with respect to the vectors k₀, k₁, and k₂ after the sorting.

The set generation unit 130 generates a set B_(n,j), of which each element is concealed, with respect to each of n=0, . . . , N−1 and each of j=1, . . . , K_(n), through secure computation with the set generation units 130 of other matrix and key generation devices 160 so that individual elements correspond to combinations of N−1 pieces of elements, which are individually selected from sets M₀, . . . , M_(N−1) other than the set M_(n), and x_(n)[j], and the elements for all of the combinations are included (S130). For example, a combination is generated as (an element of M₀, . . . , an element of M_(n−1), x_(n)[j], an element of M_(n+1), . . . , an element of M_(N−1)) so as to obtain one element. Further, a value decisively computed based on a combination (a value corresponding to a combination in a one-on-one state, in other words, a value obtained such that a different computation value is always obtained with respect to a different combination) may be set as an element. The above-mentioned “corresponding to a combination” represents inclusion of a value decisively computed from a combination other than the combination itself.

Processing of the set generation unit 130 will be described while referring to the vector k₀, the vector k₁, and the vector k₂ illustrated in FIG. 6, for example. The vector k₀ has two pieces of ∥1∥ among elements thereof and the number of each of other elements is one, so that the upper limit number m₀ in which elements are duplicated in the vector k₀ is 2. When the upper limit number in which elements are duplicated is checked in a similar manner, the upper limit number m₁ in which elements are duplicated in the vector k₁ is 2 and the upper limit number m₂ in which elements are duplicated in the vector k₂ is 3. Accordingly, set M₀={∥1∥,∥2∥}, set M₁={∥1∥,∥2∥}, and set M₂={∥1∥,∥2∥,∥3∥} are obtained. An example of processing for coupling the matrix T₀ and the matrix T₁ is first described. Since x₀[1]=∥1∥ and set M₁={∥1∥,∥2∥} hold, combinations between x₀[1] and elements of the set M₁ are (∥1∥,∥1∥) and (∥1∥,∥2∥). Accordingly, the set B_(0,1)={(∥1∥,∥1∥), (∥1∥,∥2∥)} is obtained. Further, since x₀[2]=∥2∥ and set M₁={∥1∥,∥2∥} hold, for example, combinations between x₀[2] and elements of the set M₁ are (∥2∥,∥1∥) and (∥2∥,∥2∥) and combinations between elements of the set M₀ and x₁[4] are (∥1∥,∥2∥) and (∥2∥,∥2∥). The processing may be performed in a similar manner with respect to other combinations. The case of processing for coupling the matrix T₀, the matrix T₁, and the matrix T₂ takes combinations of the three. Since x₀[1]=∥1∥, set M₁={∥1∥,∥2∥}, and set M₂={∥1∥,∥2∥,∥3∥} hold, combinations among x₀[1], elements of the set M₁, and elements of the set M₂ are (∥1∥,∥1∥,∥1∥), (∥1∥,∥1∥,∥2∥), (∥1∥,∥1∥,∥3∥), (∥1∥,∥2∥,∥1∥), (∥1∥,∥2∥,∥2∥), and (∥1∥,∥2∥,∥3∥). Accordingly, set B_(0,1)={{(∥1∥,∥1∥,∥1∥), (∥1∥,∥1∥,∥2∥), (∥1∥,∥1∥,∥3∥), (∥1∥,∥2∥,μ1∥), (∥1∥,∥2∥,∥2∥), (∥1∥,∥2∥,∥3∥)}} is obtained.

The matrix generation unit 140 generates a matrix T_(n), of which each element is concealed, with respect to each of n=0, . . . , N−1 through secure computation with the matrix generation units 140 of other matrix and key generation devices 160, so that rows identical to T_(n)[j] are included in the number equal to the number of elements of the set B_(n,j) for all of j=1, . . . , K_(n) (S140). FIG. 7 illustrates examples of the matrix T₀′, the vector k₀′, the matrix T₁′, and the vector k₁′, of which elements are concealed, in the case of coupling the matrix T₀ and the matrix T₁ illustrated in FIG. 2. In the case of processing for coupling the matrix T₀ and the matrix T₁, the number of elements of the set B_(0,1) is 2, so that two rows identical to T₀[1] are generated. In a similar manner, since the number of elements of each of the set B_(0,2), B_(0,3), B_(0,4), and B_(0,5) is 2 as well, two identical rows are generated for each set. Accordingly, the matrix T₀′ includes 10 rows which include identical rows two by two. Further, the matrix T₁′ includes 8 rows which include identical rows two by two, as well.

The key generation unit 150 generates a vector k_(n)′, of which each element is concealed, with respect to each of n=0, . . . , N−1 through secure computation with the key generation units 150 of other matrix and key generation devices 160 so that an element of the matrix T_(n)′ which corresponds to a row identical to T_(n)[j] corresponds to a combination of k_(n)[j] and an element of the set B_(n,j) and further, the elements of the set B_(n,j) are different from each other when there are a plurality of rows identical to T_(n)[j] (S150). The matrix T₀′ has ten rows and the matrix T₁′ has eight rows, so that the number of elements of the vector k₀′ is ten and the number of elements of the vector k₁′ is eight. Since both of T₀′[1] and T₀′[2] are T₀[1], k₀′[1] is (k₀[1], one element of B_(0,1)) and k₀′[2] is (k₀[1], another element of B_(0,1)). In FIG. 7, k₀′[1]=(∥1∥,(∥1∥,∥1∥)) and k₀′[2]=(∥1∥m(∥1∥,∥2∥)) hold. That is, elements (k₀′[1] and k₀′[2]) of the matrix T₀′ which correspond to a row identical to T₀[1] correspond to combinations of the k₀[1] and elements of the set B_(0,1). Further, there are a plurality of rows identical to T₀[1], so that the elements of the set B_(0,1) are selected so that the selected elements are different from each other.

To “correspond to a combination” in the description of the key generation unit 150 also represents inclusion of a value decisively computed from a combination (a value corresponding to the combination in a one-on-one state) other than the combination itself. For example, f may be set as a function for decisively computing a value based on a combination and k₀′[1]=∥f(∥1∥,(∥1∥,∥1∥))∥ may be set. Here, f denotes a function permitting secure computation.

Apparent from FIG. 7, there is no duplication of elements in either the vector k₀′ or the vector k₁′. Thus, according to the matrix and key generation device of the first embodiment, a vector which includes duplicate elements and a matrix which is a coupling object can be respectively converted into a vector which includes no duplicate elements and a matrix corresponding to the vector.

[Modification]

FIG. 8 illustrates a configuration example of a matrix coupling system and FIG. 9 illustrates a processing flow of the matrix coupling system. The matrix coupling system includes three or more matrix coupling devices 100 which are mutually connected via the network 800 and secure computation can be performed among the matrix coupling devices 100. The matrix coupling device 100 includes the matrix and key generation device 160 and a coupling unit 170. The matrix and key generation device 160 and the matrix and key generation step S160 are same as those described in the first embodiment.

In the case where there are elements common to all of the vectors k₀′, k_(NA)′, the coupling unit 170 couples corresponding rows of the matrices T₀′, . . . , T_(N−1)′ for each of the common elements to generate one row through secure computation with the coupling units 170 of other matrix coupling devices 100 so as to generate a matrix of which each element is concealed (S170). As this processing, the technique of matrix coupling described in Non-patent Literature 3 may be employed. FIG. 10 illustrates a matrix obtained by coupling the matrix T₀′ and the matrix T₁′ which are illustrated in FIG. 7. It is understood that the matrix illustrated in FIG. 10 is a matrix obtained by concealing each element of a matrix obtained by coupling the matrix T₀ and the matrix T₁ illustrated in FIG. 2. Here, a corresponding relationship among rows may be recognized in the processing up to processing for obtaining the matrices T₀′, . . . , T_(N−1)′ and the vectors k₀′, k_(N−1)′. However, if rows are replaced at random in coupling of matrices through secure computation, recognition of the corresponding relationship among rows can be prevented.

In the technique of matrix coupling of Non-patent Literature 3, when a sum of the number of records in a table of the input (a sum of the number of rows in the case of expression in a matrix form) is denoted by Q, the communication amount is O(Q·logQ). In the case of the matrix-matrix coupling system according to the present invention, when a sum of the number of rows is denoted by Q and the upper limit of the number of duplication is denoted by P, the communication amount can be set as O(PQ·logQ). In the case where there is duplication among keys, processing is increased in the order of the square of P in general. However, processing is increased in the order of the first power of P in the present invention, being able to take advantage of Non-patent Literature 3 in which the communication amount can be reduced.

Second Embodiment

The case of the secure computation has been discussed in the first embodiment, but the idea of the present invention is applicable without limiting to the secure computation. When secure computation is not used, a plurality of matrix and key generation devices do not have to be used. One matrix and key generation device can convert a vector which represents keys and includes duplicate elements and a matrix which is a coupling object respectively into a vector which represents a key and includes no duplicate elements and a matrix corresponding to the vector. Accordingly, the case where concealment is not performed will be described in the second embodiment.

In the second embodiment, it is assumed that N is an integer and 1≤N, n is an integer and 0≤n≤N−1, K_(n) is an integer and 1≤K_(n), i and j are integers, T_(n,j) is a matrix having K_(n,j) rows, k_(n,j) is a vector which includes pieces of elements, T_(n)[j] is a row on the j-th order of the matrix T_(n), k_(n)[j] is an element on the j-th order of the vector k, m is the upper limit number in which the elements of the vector k_(n) are duplicated, and matrices T₀, . . . , and vectors k₀, are inputs. Further, it is assumed that M, is a set which is composed of m_(n) pieces of elements and of which M_(n)[i]=i is satisfied.

FIG. 11 illustrates a configuration example of a matrix and key generation device according to the second embodiment, and FIG. 4 illustrates an example of a processing flow of the matrix and key generation device according to the second embodiment. A matrix and key generation device 260 includes a sort unit 210, a vector generation unit 220, a set generation unit 230, a matrix generation unit 240, and a key generation unit 250. The sort unit 210 performs sorting of orders of rows of the matrix T_(n), and sorting of orders of elements of the vector k_(n) with respect to each of n=0, . . . , N−1 in accordance with the elements of the vector k_(n) while maintaining correspondence so as to update the matrices T₀, . . . , T_(N−1) and the vectors k₀, k_(N−1) with the matrices and the vectors after the sorting (S210). The vector generation unit 220, the set generation unit 230, the matrix generation unit 240, and the key generation unit 250 perform processing with respect to the matrices T₀, . . . , T_(N−1) and the vectors k₀, . . . , k_(N−1) which are updated by the sort unit 210.

The vector generation unit 220 generates a vector x_(n) with respect to each of n=0, . . . , N−1 so that x_(n)[i]=1 and x_(n)[i]=x_(n)[i−1]+1 if k_(n)[i−1]=k_(n)[i] and x_(n)[i]=1 if k_(n)[i−1]≠k_(n)[i] with respect to 2≤i≤K_(n) (S220). The set generation unit 230 generates a set B_(n,j) with respect to each of n=0, . . . , N−1 and each of j=1, . . . , K_(n), so that individual elements correspond to combinations of N−1 pieces of elements, which are individually selected from sets M₀, M_(N−1) other than the set M_(n), and x_(n)[j], and the elements for all of the combinations are included (S230). The matrix generation unit 240 generates a matrix T_(n)′ with respect to each of n=0, . . . , N−1 so that rows identical to T_(n)[j] are included in the number equal to the number of elements of the set B_(n)for all of j=1, . . . , K_(n), (S240). The key generation unit 250 generates a vector k_(n)′ with respect to each of n=0, . . . , N−1 so that an element of the matrix T_(n)′ which corresponds to a row identical to T_(n)[j] corresponds to a combination of k_(n)[j] and an element of the set B_(n,j) and further, the elements of the set B_(n,j) are different from each other when there are a plurality of rows identical to T_(n)[j] (S250).

Each processing is different from the processing of the first embodiment only in that secure computation is not performed. Therefore, in the case where the matrices T₀ and T₁ and the vectors k₀ and k₁ illustrated in FIG. 2 are inputted, the matrices T₀′ and T₁′ and the vectors k₀′ and k₁′ to be outputted are matrices and vectors in which each element illustrated in FIG. 7 is not concealed. Thus, according to the matrix and key generation device of the second embodiment, a vector which includes duplicate elements and a matrix which is a coupling object can be respectively converted into a vector which includes no duplicate elements and a matrix corresponding to the vector.

[Modification]

A generic concept of the second embodiment will be derived in this modification. In the present modification, the sort unit 210 is omitted and the vector generation unit 220 is replaced with a vector generation unit 220′. Further, in the present modification, M_(n) is not limited to a set of M_(n)[i]=i but M_(n) is a set composed of m_(n) pieces of elements which are different from each other and M_(n)[i] is an element on the i-th order of the set M_(n). The functional configuration of a matrix and key generation device according to the present modification is illustrated in FIG. 11 and a processing flow is illustrated in FIG. 4. In the present modification, the vector generation unit 220′, the set generation unit 230, the matrix generation unit 240, and the key generation unit 250 perform processing with respect to the matrices T₀, . . . , T_(N−1) and the vectors k₀, . . . , k_(N−1) which are inputted (the sort step S210 is not performed).

The vector generation unit 220′ generates a vector x_(n), of which the number of pieces of elements is K_(n), and each of the elements is an element of the set M_(n), with respect to each of n=0, . . . , N−1 so that x_(n)[i]≠x_(n)[j] if k[i]=k_(n)[j] at i≠j. The processing of the vector generation unit 220 in the second modification is one processing which satisfies a condition of processing of the vector generation unit 220′ which is applicable when the vector k_(n) is preliminarily subjected to sorting. Accordingly, the invention of the present modification is the general concept of the invention of the second embodiment.

Since whether or not sorting is performed does not exert any influence on processing of the set generation unit 230, the matrix generation unit 240, and the key generation unit 250 when the vector x_(n) is generated as described above, the matrix and key generation device of the present modification is also capable of converting a vector which includes duplicate elements and a matrix which is a coupling object respectively into a vector which includes no duplicate elements and a matrix corresponding to the vector.

[Program, Recording Medium]

The above-described various types of processing may be executed not only in a time-series manner in accordance with the description but also in a parallel manner or an independent manner, depending on processing capability of the device which executes the processing or as necessary. Further, it is indisputable that alterations can be arbitrarily made without departing from the intent of the present invention.

In a case where the above-described configuration is implemented by a computer, processing contents of functions which should be obtained by respective devices are described by a program. By executing this program by a computer, the above-described processing functions are implemented on the computer.

The program in which the processing contents are described can be recorded in a computer-readable recording medium. Any recording medium such as a magnetic recording device, an optical disk, a magneto-optical recording medium, and a semiconductor memory may be employed as the computer-readable recording medium.

Moreover, this program is distributed by selling, transferring, or lending a portable recording medium such as a DVD and a CD-ROM in which the program is recorded, for example. Further, this program may be distributed such that this program is preliminarily stored in a storage device of a server computer and is transferred from the server computer to other computers through the network.

A computer which executes such program once stores the program which is recorded in a portable recording medium or the program transferred from the server computer in a storage device thereof, for example. Then, at the time of execution of processing, this computer reads the program which is stored in a recording medium thereof and executes processing in accordance with the program which is read. Moreover, as another executing configuration of this program, a computer may directly read the program from a portable recording medium so as to execute processing in accordance with the program. Furthermore, every time the program is transferred to the computer from the server computer, the computer may sequentially execute the processing in accordance with the received program. Alternatively, the above-described processing may be executed by so-called application service provider (ASP) type service by which a processing function is implemented only by an executing instruction of processing and result acquisition, without transferring the program to the computer from the server computer. Here, it should be noted that the program according to this executing configuration includes information which is provided for processing performed by an electronic calculator and is equivalent to the program (such as data which is not a direct instruction to the computer but has a property specifying the processing of the computer).

In this configuration, the devices are assumed to be configured as a result of a predetermined program executed on a computer. However, at least part of these processing contents may be implemented on the hardware.

INDUSTRIAL APPLICABILITY

The present invention is applicable to statistical processing and analysis of data using a computer.

DESCRIPTION OF REFERENCE NUMERALS

-   100 matrix coupling device -   110, 210 sort unit -   120, 220 vector generation unit -   130, 230 set generation unit -   140, 240 matrix generation unit -   150, 250 key generation unit -   160, 260 matrix and key generation device -   170 coupling unit -   190 recording unit -   800 network 

1. A matrix and key generation device in which N is an integer which is 1 or larger, n is an integer which is between 0 and N−1 inclusive, K_(n) is an integer which is 1 or larger, i and j are integers, T_(n) is a matrix having K_(n) rows, k_(n) is a vector which includes K_(n) pieces of elements, T_(n)[j] is a row on a j-th order of the matrix T_(n), k_(n)[j] is an element on a j-th order of the vector k_(n), m_(n) is an upper limit number in which the elements of the vector k_(n) are duplicated, M_(n) is a set composed of m_(n) pieces of elements which are different from each other, M_(n)[i] is an element on an i-th order of the set M_(n), the element on the j-th order of the vector k_(n) is an element corresponding to a row on the j-th order of the matrix T_(n), and the matrices T₀, . . . , T_(N−1) and the vectors k₀, k_(N−1) are inputs, the matrix and key generation device comprising: a vector generation unit which generates a vector x, of which a number of pieces of elements is K_(n) and each of the elements is an element of the set M_(n), with respect to each of n=0, . . . , N−1 so that x_(n)[i]±x,[j] if k_(n)[i]=k_(n)[j] at i≠j; a set generation unit which generates a set B_(ni) with respect to each of n=0, . . . , N−1 and each of j=1, K_(n) so that individual elements correspond to combinations of N−1 pieces of elements, the N−1 pieces of elements being individually selected from sets M₀, . . . , M_(N−1) other than the set M_(n), and x_(n)[j], and the elements for all of the combinations are included; a matrix generation unit which generates a matrix T_(n)′ with respect to each of n=0, . . . , N−1 so that rows identical to T_(n)[j] are included in a number equal to a number of elements of the set B_(n,j) for all of j=1, . . . , K_(n); and a key generation unit which generates a vector k_(n)′ with respect to each of n=0, . . . , N−1 so that an element of the matrix T_(n)′, the element corresponding to a row identical to T_(n)[j], corresponds to a combination of k_(n)[j] and an element of the set B_(n,j) and further, the elements of the set B_(n,j) are different from each other when there are a plurality of rows identical to T_(n)[j].
 2. The matrix and key generation device according to claim 1, further comprising: a sort unit which performs sorting of orders of rows of the matrix T_(n) and sorting of orders of elements of the vector k_(n), with respect to each of n=0, . . . , N−1 in accordance with the elements of the vector k_(n) while maintaining correspondence so as to update the matrices T₀, . . . , T_(N−1) and the vectors k₀, . . . , k_(N−1) with the matrix and the vector after the sorting, wherein M_(n)[i]=i holds, the vector generation unit, the set generation unit, the matrix generation unit, and the key generation unit perform processing with respect to the matrices T₀, . . . , T_(N−1) and the vectors k₀, k_(N−1) which are updated by the sort unit, and the vector generation unit generates a vector x with respect to each of n=0, . . . , N−1 so that x_(n)[1] and x_(n)[i]=x_(n)[i−1]+1 if k_(n)[i−1]=k_(n)] and x_(n)[i]=1 if k_(n)[i−1]≠k_(n)[i] with respect to 2≤i≤K_(n).
 3. A matrix and key generation device which constitutes a matrix and key generation system in which secure computation is performed among three or more matrix and key generation devices which are mutually connected via a network, in which N is an integer which is 1 or larger, n is an integer which is between 0 and N−1 inclusive, K_(n) is an integer which is 1 or larger, i and j are integers, T_(n) is a matrix having K_(n) rows, k_(n) is a vector which includes K_(n) pieces of elements, T_(n)[j] is a row on a j-th order of the matrix T, k_(n)[j] is an element on a j-th order of the vector k_(n), m_(n) is an upper limit number in which the elements of the vector k_(n) are duplicated, M_(n) is a set whose elements are 1, . . . , m_(n), and ∥ ∥ is a sign representing concealed data, the element on the j-th order of the vector k_(n) is an element corresponding to a row on the j-th order of the matrix T_(n), numbers of rows and columns of the matrix T_(n), a number of the elements of the vector k_(n), and a value m_(n) are unconcealed information, and each element of the matrix T_(n), and each of the elements of the vector k_(n), are concealed among the matrix and key generation devices, the matrix and key generation device comprising: a sort unit; a vector generation unit; a set generation unit; a matrix generation unit; and a key generation unit, wherein the sort unit performs sorting of orders of rows of the matrix T_(n) and sorting of orders of elements of the vector k_(n) with respect to each of n=0, . . . , N−1 through secure computation with sort units of other matrix and key generation devices in accordance with the elements of the vector k_(n) while maintaining correspondence, so as to update matrices T₀, . . . , T_(N−1) and vectors k₀, k_(N−1) with the matrices and the vectors after the sorting, the vector generation unit, the set generation unit, the matrix generation unit, and the key generation unit perform processing with respect to the matrices T₀, . . . , T_(N−1) and the vectors k₀, . . . , k_(N−1) which are updated by the sort unit, the vector generation unit generates a vector x_(n), of which a number of pieces of elements is K_(n) and each of the elements is concealed, with respect to each of n=0, . . . , N−1 through secure computation with vector generation units of other matrix and key generation devices so that x_(n)[1]=∥1∥ and x_(n)[i]=∥x_(n)[i−1]+1∥ if k_(n)[i−1]=k_(n)] and x_(n)[i]=∥1∥ if k_(n)[i−1]≠k_(n)[i] with respect to 2≤i≤K_(n). the set generation unit generates a set B_(n,j), of which each element is concealed, with respect to each of n=0, . . . , N−1 and each of j=1, . . . , K_(n) through secure computation with set generation units of other matrix and key generation devices so that individual elements correspond to combinations of N−1 pieces of elements, the N−1 pieces of elements being individually selected from sets M₀, . . . , M_(N−1) other than the set M_(n), and x_(n)[j], and the elements for all of the combinations are included, the matrix generation unit generates a matrix of which each element is concealed, with respect to each of n=0, . . . , N−1 through secure computation with matrix generation units of other matrix and key generation devices, so that rows identical to T_(n)[j] are included in a number equal to a number of elements of the set B_(n,j) for all of j=1, . . . K_(n), and the key generation unit generates a vector k_(n)′, of which each element is concealed, with respect to each of n=0, . . . , N−1 through secure computation with key generation units of other matrix and key generation devices, so that an element of the matrix T_(n)′, the element corresponding to a row identical to T_(n)[j], corresponds to a combination of k_(n)[j] and an element of the set B_(n,j) and further, the elements of the set B_(n,j) are different from each other when there are a plurality of rows identical to T_(n)[j].
 4. A matrix coupling device which constitutes a matrix coupling system in which secure computation is performed among three or more matrix coupling devices which are mutually connected via a network, the matrix coupling device comprising: the matrix and key generation device according to claim 3; and a coupling unit, wherein in a case where there are elements common to all of vectors k₀′, . . . , k_(N−1)′, the coupling unit couples corresponding rows of matrices T₀′, . . . , T_(N−1)′ for each of the common elements to generate one row through secure computation with coupling units of other matrix coupling devices so as to generate a matrix of which each element is concealed.
 5. A matrix and key generation system which includes three or more matrix and key generation devices according to claim
 3. 6. A matrix and key generation method for making a matrix and key generation device, the matrix and key generation device including a vector generation unit, a set generation unit, a matrix generation unit, and a key generation unit, execute processing, in which N is an integer which is 1 or larger, n is an integer which is between 0 and N−1 inclusive, K_(n) is an integer which is 1 or larger, i and j are integers, T_(n) is a matrix having K_(n) rows, k_(n) is a vector which includes K_(n) pieces of elements, T_(n)[j] is a row on a j-th order of the matrix T_(n), k_(n)[j] is an element on a j-th order of the vector k_(n), m_(n) is an upper limit number in which the elements of the vector k_(n) are duplicated, M_(n) is a set composed of m_(n) pieces of elements which are different from each other, M_(n)[i] is an element on an i-th order of the set M_(n), the element on the j-th order of the vector k_(n) is an element corresponding to a row on the j-th order of the matrix T_(n), and the matrices T₀, . . . , T_(N−1) and the vectors k₀, . . . , k_(N−1) are inputs, the matrix and key generation method for executing steps comprising: a vector generation step in which the vector generation unit generates a vector x_(n), of which a number of pieces of elements is K_(n) and each of the elements is an element of the set M_(n), with respect to each of n=0, . . . , N−1 so that x_(n)[i]≠x_(n)[j] if k_(n)[i]=k_(n)[j] at i≠j; a set generation step in which the set generation unit generates a set B_(n,j) with respect to each of n=0, . . . , N−1 and each of j=1, . . . , K_(n) so that individual elements correspond to combinations of N−1 pieces of elements, the N−1 pieces of elements being individually selected from sets M₀, . . . , M_(N−1) other than the set M_(n), and x_(n)[j], and the elements for all of the combinations are included; a matrix generation step in which the matrix generation unit generates a matrix T_(n)′ with respect to each of n=0, . . . , N−1 so that rows identical to T_(n)[j] are included in a number equal to a number of elements of the set B_(n,j) for all of j=1, . . . , K_(n); and a key generation step in which the key generation unit generates a vector k_(n)′ with respect to each of n=0, . . . , N−1 so that an element of the matrix T_(n)′, the element corresponding to a row identical to T_(n)[j], corresponds to a combination of k_(n)[j] and an element of the set B_(n,j) and further, the elements of the set B_(n,j) are different from each other when there are a plurality of rows identical to T_(n)[j].
 7. A matrix and key generation method for making a matrix and key generation system, the matrix and key generation system which includes three or more matrix and key generation devices which are mutually connected via a network and in which secure computation can be performed among the matrix and key generation devices, execute processing, in which N is an integer which is 1 or larger, n is an integer which is between 0 and N−1 inclusive, K_(n) is an integer which is 1 or larger, i and j are integers, T_(n) is a matrix having K_(n) rows, k_(n) is a vector which includes K_(n) pieces of elements, T_(n)[j] is a row on a j-th order of the matrix T_(n), k_(n)[j] is an element on a j-th order of the vector k_(n), m_(n) is an upper limit number in which the elements of the vector k_(n) are duplicated, M_(n) is a set whose elements are 1, . . . , m_(n), and ∥ ∥ is a sign representing concealed data, the element on the j-th order of the vector k_(n) is an element corresponding to a row on the j -th order of the matrix T_(n), numbers of rows and columns of the matrix T_(n), a number of the elements of the vector k_(n), and a value m_(n) are unconcealed information, each element of the matrix T_(n) and each of the elements of the vector k_(n) are concealed among the matrix and key generation devices, and each of the matrix and key generation devices includes a sort unit, a vector generation unit, a set generation unit, a matrix generation unit, and a key generation unit, the matrix and key generation method for executing steps comprising: a sort step in which the sort unit performs sorting of orders of rows of the matrix T_(n) and sorting of orders of elements of the vector k_(n) with respect to each of n=0, . . . , N−1 through secure computation with sort units of other matrix and key generation devices in accordance with the elements of the vector k_(n), while maintaining correspondence, so as to update matrices T₀, . . . , T_(N−1) and vectors k₀, . . . , k_(N−1) with the matrices and the vectors after the sorting, in which the vector generation unit, the set generation unit, the matrix generation unit, and the key generation unit perform processing with respect to the matrices T₀, . . . , T_(N−1) and the vectors k₀, . . . , k_(N−1) which are updated in the sort step; a vector generation step in which the vector generation unit of each of the matrix and key generation devices generates a vector x, of which a number of pieces of elements is K_(n) and each of the elements is concealed, with respect to each of n=0, . . . , N−1 through secure computation with vector generation units of other matrix and key generation devices so that x_(n)[1]=∥1∥ and x_(n)[i]=∥x_(n)[i−1]+1∥ if k_(n)[i−1]=k_(n)] and x_(n)[i]=∥1∥ if k_(n)[i−1]≠k_(n)[i] with respect to 2≤i≤K_(n). a set generation step in which the set generation unit of each of the matrix and key generation devices generates a set B_(n,j), of which each element is concealed, with respect to each of n=0, . . . , N−1 and each of j=1, . . . , K_(n) through secure computation with set generation units of other matrix and key generation devices so that individual elements correspond to combinations of N−1 pieces of elements, the N−1 pieces of elements being individually selected from sets M₀, . . . , M_(N−1) other than the set M, and x_(n)[j], and the elements for all of the combinations are included; a matrix generation step in which the matrix generation unit of each of the matrix and key generation devices generates a matrix T_(n)′, of which each element is concealed, with respect to each of n=0, . . . , N−1 through secure computation with matrix generation units of other matrix and key generation devices, so that rows identical to T_(n)[j] are included in a number equal to a number of elements of the set B_(n,j) for all of j=1, . . . , K_(n); and a key generation step in which the key generation unit of each of the matrix and key generation devices generates a vector k_(n)′, of which each element is concealed, with respect to each of n=0, . . . , N−1 through secure computation with key generation units of other matrix and key generation devices, so that an element of the matrix T_(n)′, the element corresponding to a row identical to T_(n)[j], corresponds to a combination of k_(n)[j] and an element of the set B_(n,j) and further, the elements of the set B_(n,j) are different from each other when there are a plurality of rows identical to T_(n)[j].
 8. (canceled) 